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Inverse Problems and Mathematical Modeling: Analysis, Existence, Uniqueness, and Instability

This article explores various aspects of inverse problems and mathematical modeling, including the analysis of data, discrete and continuous data, sources of noise, mathematical models, linear and nonlinear models, physical laws determining the operator G, issues of existence, uniqueness, and instability in model estimation.

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Inverse Problems and Mathematical Modeling: Analysis, Existence, Uniqueness, and Instability

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  1. G(m)=d mathematical model d data m model G operator d=G(mtrue)+ = dtrue +  Forward problem: find d given m Inverse problem (discrete parameter estimation): find m given d Discrete linear inverse problem: Gm=d

  2. Continuous inverse problem:  g(s,x)m(x)dx=d(s) g is the kernel Convolution equation:  g(s-x)m(x)dx=d(s) b a b a

  3. Example: linear regression for ballistic trajectory y(t)=m1+m2t-0.5m3t2 m1 initial altitude, m2 initial vertical velocity, m3 effective gravitational acceleration o • 1 t1 -0.5t12 m1 y1 • 1 t2 -0.5t22 m2 y2 • 1 t3 -0.5t32 m3 =y3 • ….. . • 1 tm -0.5tm2 ym Y(t) o o o t

  4. Earthquake location m=[x ] G(m)=t ti=||S.,i-x||2/c+ (arrival time of wave at station i) Nonlinear problem!  

  5. j Traveltime tomography T = ∫ 1/v(s)ds = ∫u(s)ds Tj = ∑ Gij ui j-th ray i=1

  6. Gravity d(s) h (x) ∞ -∞ ∞ -∞ d(s) =G  h m(x) dx / [(x-s)2+h2]3/2 =  g(x-s) m(x) dx d(s) h(x)  ∞ -∞ d(s) = G  m(x)  dx / [(x-s)2+m(x)2]3/2 nonlinear in m(x)

  7. Existence: maybe no model that fits data (bad model, noisy data)Uniqueness: maybe several (infinite?) number of models that fit dataInstability: small change in data leading to large change in estimateAnalysis:What are the data?Discrete or continuous data?Sources of noise?What is the mathematical model?Discrete or continuous model?What physical laws determine G?Is G linear or nonlinear?Any issues of existence, uniqueness, or instability?

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